Use The Data File For Question 1 To Answer This Question

use The Data File Data For Question 1 To Answer This Ques

Question 1: Use the data file “data for question 1” to answer this question. In this question, the variable of interest is the daily production of central processing units (CPUs) at a chip manufacturing factory. This data file is a sample of 500 days of CPU production. a) Construct the histogram for this data. Manipulate the number of bins and class dimensions to produce a histogram that provides a good idea of the distribution of this data. Choose the cutpoint option. b) Once you have constructed the histogram, construct the absolute and relative frequency distribution for this data. c) Provide the five-number summary. What do these numbers mean? d) Provide the mean and standard deviation for this data. What do these numbers mean? e) Use the empirical rule to construct three intervals: the interval that contains the middle 68% of the data, the interval that contains the middle 95% of the data, and the interval that contains the middle 99.73% of the data.

Question 2: There is no data set for this problem. In order to be hired as an actuary, an applicant must take and successfully pass an exam prepared by the Actuarial Society of America. The initial exam concerns mathematics and mathematical statistics. The scores from this exam are normally distributed (i.e., the histogram of exam scores is bell-shaped and symmetric). Furthermore, the mean is 60, and the standard deviation is 12. Use this information to answer the following questions: a) What is the probability that a randomly selected individual who took the exam has a score less than 70? b) What is the probability that a randomly selected individual who took the exam has a score greater than 53? c) What is the probability that a randomly selected individual who took the exam has a score between 53 and 70? d) What is the score that has 20% of the scores below it (or, 80% of the scores above it)? e) What is the score that has 95% of the scores below it (or, 5% of the scores above it)?

Paper For Above instruction

The analysis of production data and the application of statistical principles in risk assessment are fundamental in manufacturing and actuarial sciences. This paper explores two core statistical tasks: the analysis of CPU production data within a manufacturing context and the evaluation of exam score distributions critical for actuarial certification processes. Using a combination of descriptive statistics, probability, and empirical rule applications, insights into data distribution, variability, and percentile calculations will be presented.

Part 1: CPU Production Data Analysis

The first task involves analyzing a dataset representing 500 days of CPU production at a chip manufacturing factory. The goal is to understand the distribution, variability, and central tendency of daily CPU output. Constructing an effective histogram is crucial for visualizing the data's distribution. By experimenting with bin widths and cutpoints, an optimal histogram reveals whether the data is normal, skewed, or multimodal, guiding subsequent statistical analysis.

Following the histogram, calculating the absolute and relative frequencies allows quantification of how often specific production ranges occur. These distributions help identify common production levels and outliers, essential for assessing process stability and capacity planning.

The five-number summary — comprising the minimum, first quartile (Q1), median, third quartile (Q3), and maximum — succinctly summarizes the data’s spread and center. These numbers indicate the range of production, the median daily production, and the interquartile range (IQR), reflecting variability and skewness. A large gap between Q1 and Q3 suggests dispersion, while the position of the median relative to the min and max indicates skewness.

Calculating the mean and standard deviation provides measures of central tendency and dispersion. The mean indicates the average daily production, while the standard deviation quantifies variability around this mean. These metrics are essential for control charting, capacity planning, and process improvement initiatives.

Applying the empirical rule—assuming an approximately normal distribution—enables constructing three key intervals:

• Approximately 68% of production data falls within one standard deviation of the mean.
• Approximately 95% within two standard deviations.
• Almost all (

These intervals assist in detecting unusual production days that fall outside expected ranges, supporting quality control efforts.

Part 2: Exam Score Distribution and Probabilities

The second task deals with an actuarial exam whose scores follow a normal distribution with a mean of 60 and a standard deviation of 12. This problem illustrates probability calculations pertinent to exam performance, crucial for assessing candidate performance and setting scoring thresholds.

Using the properties of the normal distribution, probabilities of scoring below or above certain thresholds are computed through z-scores and cumulative distribution functions (CDF). For instance, the probability that a candidate scores less than 70 involves calculating the z-score: (70 - 60)/12 ≈ 0.83, and then consulting the standard normal distribution table or using statistical software to find the corresponding probability.

Similarly, for scores greater than 53, the z-score is (53 - 60)/12 ≈ -0.58, and the probability is 1 minus the CDF value at that z-score. For scores between 53 and 70, the difference between the probabilities at those two z-scores provides the likelihood of falling within this range.

Inverse normal calculations identify the scores associated with specific percentiles. For example, the score below which 20% of scores fall (the 20th percentile) corresponds to a z-score of approximately -0.84, resulting in a score of 60 + (-0.84)(12) ≈ 50.08. Likewise, the 95th percentile score corresponds to a z-score of approximately 1.645, yielding 60 + 1.645(12) ≈ 78.74.

These percentile scores and probabilities are fundamental in setting cutoffs, grading policies, and understanding candidate performance distributions.

Conclusion

The application of descriptive and inferential statistics in manufacturing and actuarial contexts provides valuable insights into process control and performance evaluation. Visual tools like histograms, descriptive measures like the five-number summary, and probability calculations grounded in the empirical rule and normal distribution theory facilitate informed decision-making. As illustrated, understanding data distribution and variability is crucial across industries for quality assurance and risk assessment.

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